We establish, for $1<p<\\infty$, higher order $\\mathcal{S}^{p}$-differentiability results of the function $\\varphi \\colon t\\in \\mathbb{R} \\mapsto f(A+tK) - f(A)$ selfadjoint operators $A$ and $K$ on a separable Hilbert space $\\mathcal{H}$ with element Schatten class $\\mathcal{S}^{p}(\\mathcal{H})$ $f$ $n$-times differentiable $\\mathbb{R}$. prove that if either $f^{(n)}$ are bound...

Let \(\psi\) be univalent, we introduce a newly defined class of analytic functions, given by $$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ (0)=0 \right\} . \end{aligned}$$Note that functions in this may not univalent. Further, establish the growth theorem with some geometrical conditions on and obtain Koebe domain related shar...

Given a family of graphs $\mathcal{F}$, we prove that the normalized edit distance any given graph $\Gamma$ to being induced $\mathcal{F}$-free is estimable with query complexity depends only on bounds Frieze--Kannan regularity lemma and removal for $\mathcal{F}$.

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...

We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schr\"odinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, usual $N$-body Hamiltonians with Coulomb-type singular are covered by our result: that case, weight is $\delta_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, wh...

Let $\mathbf{G}$ be a connected reductive group defined over locally compact non-archimedean field $F$, let $\mathbf{P}$ parabolic subgroup with Levi $\mathbf{M}$ and compatible pro-$p$ Iwahori of $G := \mathbf{G}(F)$. $R$ commutative unital ring. We introduce the Iwahori--Hecke $R$-algebra $\mathcal{H}_R(P)$ $P \mathbf{P}(F)$ construct two morphisms $\Theta^P_M\colon \mathcal{H}_R(P)\to \mathc...